The upper bound estimate of the number of integer points on elliptic curves
Journal of Inequalities and Applications volume 2014, Article number: 104 (2014)
Let p be a fixed prime and r be a fixed positive integer. Further let denote the number of pairs of integer points on the elliptic curve with . Using some properties of Diophantine equations, we give a sharper upper bound estimate for . That is, we prove that , except with , where s is a nonnegative integer.
Let ℤ, ℕ be the sets of all integers and positive integers, respectively. Let p be a fixed prime and k be a fixed positive integer. Recently, the integer points on the elliptic curve
where r is a positive integer.
An integer point on (1.2) is called trivial or non-trivial according to whether or not. Obviously, (1.2) has only the trivial integer point . Notice that if is a non-trivial integer point on (1.2), then is also. Therefore, along with are called by a pair of non-trivial integer points and denoted by , where . For any positive integer n, let
Using some properties of Diophantine equations, we give a sharper upper bound estimate for , the number of pairs of non-trivial integer points on (1.2). That is, we shall prove the following results.
Theorem 1.1 All non-trivial integer points on (1.2) are given as follows.
, , , where m, s are nonnegative integers.
, , , where s is a nonnegative integer, is a solution of the equation(1.5)
Theorem 1.2 Let p be an odd prime, r be a positive integer. Then for any nonnegative integer s, we have , except with . Moreover, if , then , except with .
Lemma 2.1 ([, Theorem 244])
Every solution of the equation
can be expressed as , where n is a positive integer.
Lemma 2.2 If , then , where m is a nonnegative integer.
Proof Assume that n has an odd divisor d with . Then we have either and or and . Therefore, since p is a prime, it is impossible. Thus, we get . The lemma is proved. □
Any fixed positive integer a can be uniquely expressed as , where b, c are positive integers with b is square free. Then b is called the quadratfrei of a and denoted by .
Lemma 2.3 For any positive integer m, we have .
Proof By (1.3) and (1.4), we get
Since , where is the Legendre symbol, we see from (2.3) that for . Therefore, since , by (2.2), we obtain . It implies that . The lemma is proved. □
Let D be a non-square positive integer. It is a well known fact that if the equation
has solutions , then it has a unique solution such that , where through all solutions of (2.4). For any odd positive integer l, let
Then () are all solutions of (2.4).
Lemma 2.4 ([, Theorem 1])
has at most one solution . Moreover, if the solution exists, then , where .
Lemma 2.5 ([, Theorem 3])
If , then (2.5) has no solutions .
Lemma 2.6 If , where m is a positive integer with , then (1.5) has no solutions .
Proof Since with , by (2.3), we have
We see from (2.6) that the equation
has solution and its fundamental solution is . Further, since , by Lemma 2.3, we have . Hence, we get . Therefore, by Lemma 2.5, the lemma is proved. □
Lemma 2.7 ()
has no solutions .
Lemma 2.8 The equation
has only the solutions , where m is a nonnegative integer.
Proof Assume that is a solution of (2.9). If , since , then we have , , , and . But since is not a square, it is impossible.
If p is an odd prime, then we have , and by (2.9), we get , ,
By Lemma 2.7, we get from (2.11) that and
Further, applying Lemma 2.1 to (2.12) yields
Further, by Lemma 2.2, we see from the first equality of (2.13) that . Thus, by (2.10) and (2.13), the lemma is proved. □
3 Proof of Theorem 1.1
Assume that is a pair of non-trivial integer points on (1.2). Since , we have and x can be expressed as
Substituting (3.1) into (1.2) yields
We first consider the case that . By (3.2), we have
Since , we have and . Hence by (3.3), we get
whence we obtain
Applying Lemma 2.8 to (3.5) yields
Therefore, by (3.1), (3.4), and (3.6), the integer points of type (i) are given.
We next consider the case that . Then we have
Since , and is not a square, we see from (3.7) that
By (3.8), we get
It implies that is a solution of (1.5). Therefore, by (3.1) and (3.8), we obtain the integer points of type (ii).
We finally consider the case that . Then we have
Since and , we see from (3.10) that is a square, a contradiction.
To sum up, the theorem is proved.
4 Proof of Theorem 1.2
By (2.3), if with , then . Therefore, by Theorem 1.1, if , then (1.2) has only the non-trivial integer point
It implies that the theorem is true for .
For , let and denote the number of pairs of non-trivial integer points of types (i) and (ii) in Theorem 1.1, respectively. Obviously, we have
and . By Lemma 2.4, we get . Hence, by (4.2), we have for . Since and (1.5) has the solution for , by Theorem 1.1, we get
and . However, by Lemma 2.6, if with , then . Therefore, by (4.2), if , then , except with . The theorem is proved.
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The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the P.E.D. (2013JK0573) and N.S.F. (11371291) of P.R. China.
The authors declare that they have no competing interests.
JZ obtained the theorems and completed the proof. XL corrected and improved the final version. Both authors read and approved the final manuscript.
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Zhang, J., Li, X. The upper bound estimate of the number of integer points on elliptic curves . J Inequal Appl 2014, 104 (2014). https://doi.org/10.1186/1029-242X-2014-104